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Properties

Label 12.0.49705157780...7449.3
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 7^{10}\cdot 17^{6}$
Root discriminant $36.14$
Ramified primes $3, 7, 17$
Class number $130$
Class group $[130]$
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41581, 20959, 10628, 9649, 13879, -300, 4745, -288, 691, -31, 44, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 44*x^10 - 31*x^9 + 691*x^8 - 288*x^7 + 4745*x^6 - 300*x^5 + 13879*x^4 + 9649*x^3 + 10628*x^2 + 20959*x + 41581)
 
gp: K = bnfinit(x^12 - x^11 + 44*x^10 - 31*x^9 + 691*x^8 - 288*x^7 + 4745*x^6 - 300*x^5 + 13879*x^4 + 9649*x^3 + 10628*x^2 + 20959*x + 41581, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 44 x^{10} - 31 x^{9} + 691 x^{8} - 288 x^{7} + 4745 x^{6} - 300 x^{5} + 13879 x^{4} + 9649 x^{3} + 10628 x^{2} + 20959 x + 41581 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4970515778063137449=3^{6}\cdot 7^{10}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.14$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(357=3\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{357}(256,·)$, $\chi_{357}(1,·)$, $\chi_{357}(290,·)$, $\chi_{357}(220,·)$, $\chi_{357}(205,·)$, $\chi_{357}(271,·)$, $\chi_{357}(305,·)$, $\chi_{357}(50,·)$, $\chi_{357}(341,·)$, $\chi_{357}(118,·)$, $\chi_{357}(188,·)$, $\chi_{357}(254,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{181} a^{7} + \frac{28}{181} a^{5} + \frac{43}{181} a^{3} + \frac{86}{181} a + \frac{79}{181}$, $\frac{1}{181} a^{8} + \frac{28}{181} a^{6} + \frac{43}{181} a^{4} + \frac{86}{181} a^{2} + \frac{79}{181} a$, $\frac{1}{905} a^{9} + \frac{1}{905} a^{7} - \frac{1}{5} a^{6} + \frac{192}{905} a^{5} + \frac{1}{5} a^{4} + \frac{11}{905} a^{3} + \frac{441}{905} a^{2} + \frac{31}{905} a + \frac{401}{905}$, $\frac{1}{91909085} a^{10} + \frac{12312}{91909085} a^{9} - \frac{43999}{91909085} a^{8} - \frac{240709}{91909085} a^{7} + \frac{8642838}{18381817} a^{6} + \frac{2425504}{18381817} a^{5} - \frac{12286457}{91909085} a^{4} - \frac{26786422}{91909085} a^{3} - \frac{18053512}{91909085} a^{2} + \frac{27373093}{91909085} a - \frac{34760818}{91909085}$, $\frac{1}{91909085} a^{11} - \frac{4742}{91909085} a^{9} + \frac{176169}{91909085} a^{8} + \frac{25299}{91909085} a^{7} - \frac{25799801}{91909085} a^{6} - \frac{3320391}{18381817} a^{5} - \frac{5986622}{91909085} a^{4} - \frac{42373302}{91909085} a^{3} - \frac{23020182}{91909085} a^{2} + \frac{25353752}{91909085} a + \frac{21914662}{91909085}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{130}$, which has order $130$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 140.798796005 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-119}) \), \(\Q(\sqrt{-51}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{21}, \sqrt{-51})\), \(\Q(\zeta_{21})^+\), 6.0.82572791.1, 6.0.318495051.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$17$17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$